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# End-Term Examination

First Semester [B.Tech] – December 2003

 Paper Code : ETMA – 101 Subject : Applied Mathematics - I
 Time : 3 Hours Maximum Marks : 75
 Note : Attempt any 5 questions in all. At least two questions from each section. All questions carry equal marks.

SECTION-A

 Q1 ( a ) State and prove Lagrange's Mean value theorem and verify it for the function f(x) = sin x + cos x in [0,π/2]. 5 ( b ) If y = (sin-1 x)2, prove that (1 – x2) y2 - xy1 = 2 and also show that (1 – x2) yn+2 – (2n + 1) xyn+1 – n2yn = 0 5 ( c ) If ρ1 and ρ2 are the radii of curvature at the extremities of two conjugate diameters of the ellipsex2/a2 + y2/b2 = 1, then prove that (ρ12/3 + ρ22/3)(ab)2/3 = a2 + b2 5

 Q2 ( a ) Use Taylor's theorem to evaluate √10 correct to four significant figures. 5 ( b ) Evaluate: 5 ( c ) Determine the asymptotes of 4x3 - 3xy2 - y3 + 2x2 - xy - y2 - 1 = 0 5

 Q3 ( a ) Show that(i) (ii) 5 ( b ) Test for the convergence of the following series : - x2 + (22 / 3.4) x4 + (22 .42 / 3.4.5.6) x6 + (22 .42 .62 / 3.4.5.6.7.8) x8 + ..... log(2/1) – log(3/2) + log(4/3) – log(5/4) +..... 5 ( c ) The velocity v(km/min) of a moped, which starts from rest is given at fixed intervals of time. Estimate approximately the distance covered in 20 minutes by applying Simpson's 1/3 rule. 5 t 2 4 6 8 10 12 14 16 18 20 v 10 18 25 29 32 20 11 5 2 0

 Q4 ( a ) Determine the length of the loop of the curve 9y2 = (x-3)(x-6)2 5 ( b ) Find the area common to the cardiode r = a(1+cos θ) and the circle r = (3/2)a and also the area of the remainder of the cardiode. 5 ( c ) If the curve r = a+b cos θ (a>b) revolves about the initial line, show that the volume generated is (4/3) π a (a2 +b2 ) 5

SECTION-B

 Q5 ( a ) If v = (x2 +y2 +z2 )-1/2 show that 5 ( b ) If prove that:(i) (ii) 5 ( c ) Expand sin(xy) in powers of (x-1) and (y-π/2) up to and including second degree terms. 5

 Q6 ( a ) Find the volume of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1 5 ( b ) Sketch the region of integration and evaluate: 5 ( c ) Changing into polar coordinates and hence evaluate: 5

 Q7 ( a ) Solve:( i ) ( ii ) 5 ( b ) Solve : ( 2xy cos x2 – 2xy +1 )dx + ( sin2 - x2 )dy = 0 5 ( c ) Solve : 5

 Q8 ( a ) If u1=(x2 x3)/x1 , u2=(x1 x3)/x2 , u3=(x1 x2)/x3 , then evaluate J ( u1, u2, u3). 5 ( b ) Solve : 5 ( c ) Solve : 5